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Operator positivity and analytic models of commuting tuples of operators. (English) Zbl 1385.47007
Authors’ abstract: We study analytic models of operators of class \(C_{\cdot 0}\) with natural positivity assumptions. In particular, we prove that for an \(m\)-hypercontraction \(T \in C_{\cdot 0}\) on a Hilbert space \(\mathcal H\), there exist Hilbert spaces \(\mathcal E\) and \(\mathcal E_*\) and a partially isometric multiplier \(\theta \in \mathcal M(H^2(\mathcal E), A^2_m(\mathcal E_*))\) such that \[ \mathcal H \mathcal Q_{{\theta}} = A^2_m(\mathcal E_*) \theta H^2(\mathcal E) \text{ and } T P_{\mathcal Q_\theta} M_z|_{\mathcal Q_\theta}, \] where \(A^2_m(\mathcal E_*)\) is the \(\mathcal E_*\)-valued weighted Bergman space and \(H^2(\mathcal E)\) is the \(\mathcal E\)-valued Hardy space over the unit disc \(\mathbb{D}\). We then proceed to study analytic models for doubly commuting \(n\)-tuples of operators and investigate their applications to joint shift co-invariant subspaces of reproducing kernel Hilbert spaces over the polydisc. In particular, we completely analyze doubly commuting quotient modules of a large class of reproducing kernel Hilbert modules, in the sense of J. Arazy and M. Engliš [Trans. Am. Math. Soc. 355, No. 2, 837–864 (2003; Zbl 1060.47013)], over the unit polydisc \(\mathbb{D}^n\).

MSC:
47A45 Canonical models for contractions and nonselfadjoint linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A15 Invariant subspaces of linear operators
47A20 Dilations, extensions, compressions of linear operators
47A80 Tensor products of linear operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B38 Linear operators on function spaces (general)
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References:
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